Lanczos method for dynamic analysis of damped structural systems

Abstract

AbstractIn this paper we extend the Lanczos algorithm for the dynamic analysis of structures7 to systems with general matrix coefficients. The equations of dynamic equilibrium are first transformed to a system of first order differential equations. Then the unsymmetric Lanczos method is used to generate two sets of vectors. These vectors are used in a method of weighted residuals to reduce the equations of motion to a small unsymmetric tridiagonal system.The algorithm is further simplified for systems of equations with symmetric matrices. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is remarkably close to that in Reference 7, but generalized to the case with indefinite matrix coefficients. This simplification eliminates one of the sets of vectors generated by the unsymmetric Lanczos method and results in a symmetric tridiagonal, but indefinite, system. We identify the difficulties that may arise when this implementation is applied to problems with symmetric indefinite matrices such as vibration of structures with velocity feedback control forces which lead to symmetric damping matrices.This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.